Efficiency & Leakage in non-circular solenoid cores

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LowEParticle
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Part of a new product we're designing needs a small iron-core D.C. electromagnet for an actuator. I designed the electromagnet's core to be circular: about 4.5mm in diameter with 200 Ampere-Turns in the coil. An axis-symmetric FEA showed the iron core to be near saturation, and producing a sufficient B field to do the work we need done.

Some other parts on the project need more space, and so I need to look at shrinking the cross-sectional space taken by the electromagnet, but only in one axis: the other axis of the cross section has room to spare, so I thought I'd simply make the iron core's cross section ellipsoid, or possibly even a rectangle with rounded corners. The third axis (axial length of the iron core) would remain the same. The shape chosen for the new cross-section would be calculated to have the same area as the original circular core, thus producing a core with the same number of magnetic domains and hopefully similar response to the original circular core.

I can't do an FEA: I'm limited to the axis-symmetic case which excludes both ellipsoid and rounded-rectangle forms. I am worried that either:

A) the new-shape core will either be significantly less efficient than the circular core, or
B) the B field leakage will increase a lot when the symmetry of the circular core is no longer present.

I would appreciate any help offered; thank you very much!
David
 
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Usually the supplier of magnetic cores will have design information or will at least supply you with some samples to do your own evaluation.

Get a few cores of different shapes and try them out.
 
That's a very practical idea; thanks very much!
David